,96 KINETICS OF A PARTICLE. [179. 



particle may leave the circle at some point of its upper half. 

 Again, the one-sided constraint may be such that OP>1, as 

 when the particle runs in a groove cut on the rim of a disc ; 

 in this case the particle can of course only move on the upper 

 half of the circle. 



179. Exercises. 



(1) For h = l, equation (10) can be integrated in finite terms. 

 Show that in this limiting case the particle approaches the highest point 

 B of the circle asymptotically, reaching it only in an infinite time. 



(2) Derive the equations of motion for the problem of the simple 

 pendulum (Art. 175) from the general equations of Arts. 172, 173. 



(3) For == 60, /= i ft., 27 = 9 ft. per second, show that the par- 

 ticle will leave the circle very nearly at the point $ 1 = 120, if the con- 

 straint be such that OP<1 (Art. 178). 



(4) For z'o= 10 ft. per second, everything else being as in Ex. (3), 

 show that the particle will leave the circle at the point ^ = 134!, 

 nearly. 



(5) A particle, subject to gravity and constrained to the inside 

 of a vertical circle (OP^_/), makes complete revolutions. Show that 

 it cannot leave the circle at any point, if \h > /; and that it will leave 

 the circle at the point for which cos = - f h/l, if f h < I. 



(6) In the experiment of swinging in a vertical circle a glass contain- 

 ing water, and suspended by means of a string, if the string be 2 ft. long, 

 what must be the velocity at the lowest point if the experiment is to 

 succeed? 



(7) A particle subject to gravity moves on the outside of a vertical 

 circle; determine where it will leave the circle: (a) if MN (Fig. 25) ' 

 intersects the circle ; (b) if MN touches the circle ; (c) if MN does 

 not meet the circle. 



(8) A particle subject to gravity is compelled to move on any vertir 

 cal curve z =/(#) without friction. Show that the velocity at any point 

 is v = ^/2gz (comp. Art. 176) if the horizontal axis of x be taken at 

 height above the initial point equal to the " height due to the initi; 

 velocity," i.c. v/2g. 



